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The concept of a rational arithmetical function originates from R. Vaidyanathaswamy (1931).
== Busche-Ramanujan identities ==
A multiplicative function <math>f</math> is said to be specially multiplicative
if there is a completely multiplicative function <math>f_A</math> such that
:<math>
f(m) f(n) = \sum_{d\mid (m,n)} f(mn/d^2) f_A(d)
</math>
for all positive integers <math>m</math> and <math>n</math>, or equivalently
:<math>
f(mn) = \sum_{d\mid (m,n)} f(m/d) f(n/d) \mu(d) f_A(d)
</math>
for all positive integers <math>m</math> and <math>n</math>, where <math>\mu</math> is the Möbius function.
These are known as Busche-Ramanujan identities.
In 1906, E. Busche stated the identity
:<math>
\sigma_k(m) \sigma_k(n) = \sum_{d\mid (m,n)} \sigma_k(mn/d^2) d^k,
</math>
and, in 1915, S. Ramanujan gave the inverse form
:<math>
\sigma_k(mn) = \sum_{d\mid (m,n)} \sigma_k(m/d) \sigma_k(n/d) \mu(d) d^k
</math>
for <math>k=0</math>. S. Chowla gave the inverse form for general <math>k</math> in 1929, see P. J. McCarthy (1986).
It is known that quadratic functions <math>f=g_1\ast g_2</math> satisfy the Busche-Ramanujan identities with <math>f_A=g_1g_2</math>. In fact, quadratic functions are exactly the same as specially multiplicative functions. Totients satisfy a restricted Busche-Ramanujan identity. For further details, see R. Vaidyanathaswamy (1931).
==Multiplicative function over {{math|''F''<sub>''q''</sub>[''X'']}}==
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==References==
* See chapter 2 of {{Apostol IANT}}
* P. J. McCarthy, Introduction to Arithmetical Functions, Universitext. New York: Springer-Verlag, 1986.
* {{cite journal |title=Efficient estimation of a multivariate multiplicative volatility model |journal=Journal of Econometrics |date=2010 |volume=159 |issue=1 |pages=55–73 |doi=10.1016/j.jeconom.2010.04.007 |s2cid=54812323 |url=http://sticerd.lse.ac.uk/dps/em/em541.pdf |last1=Hafner |first1=Christian M. |last2=Linton |first2=Oliver }}
*{{cite journal
|author=P. Haukkanen
|title=Some characterizations of specially multiplicative functions
|journal=Int. J. Math. Math. Sci.
|volume=37
|pages=2335-2344
|year=2003
|url=https://www.emis.de/journals/HOA/IJMMS/Volume2003_37/515979.abs.html
}}
*{{cite journal
|author=P. Haukkanen
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|pages=49-53
|year=1966 }}
*{{cite journal
|author=L. Tóth
|title=Two generalizations of the Busche-Ramanujan identities
|journal=International Journal of Number Theory
|volume=9
|pages=1301-1311
|year=2013 }}
*{{cite journal
|author=R. Vaidyanathaswamy
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|doi=10.1090/S0002-9947-1931-1501607-1
|doi-access=free }}
*S. Ramanujan, Some formulae in the analytic theory of numbers. Messenger 45 (1915), 81--84.
*E. Busche, Lösung einer Aufgabe über Teileranzahlen. Mitt. Math. Ges. Hamb. 4, 229--237 (1906)
==External links==
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